{
  "schema_version": 1,
  "problem_number": "OWR-14299088-013",
  "title": "A Negative Answer to a Question of Basu and Perrucci on Two Multi-Affine Polynomials",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "A real polynomial is multi-affine if it has degree at most one in each variable. Basu and Perrucci proved that the real zero set of one multi-affine polynomial of degree d in R^n has at most 2^(d-1) connected components, independently of n. They asked whether the number of connected components of the common real zero set of two multi-affine polynomials of degree at most d is bounded in terms of d alone; the question was posed again in Oberwolfach Report 9/2025. We show that the answer is no. For m >= 2, explicit multi-affine polynomials of degrees 2 and 4 in 2m+1 variables have a common real zero set homeomorphic to a product of m hyperbolas, with exactly 2^m connected components. A sum-of-squares identity gives a short exact certificate. The construction also combines multi-affine polynomials in disjoint variable sets into a pair whose zero set is homeomorphic to the product of their zero sets. The negative answer persists inside boxes and, as a lower bound, inside any set with nonempty interior.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.AG"
  ],
  "keywords": [
    "OWR-14299088-013",
    "multi-affine polynomials",
    "connected components",
    "real algebraic sets",
    "counterexample",
    "Oberwolfach Reports",
    "math.AG"
  ],
  "manuscript_version_date": "2026-09-30",
  "publication_date": "2026-09-30",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-30",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-14299088-013/",
  "pdf_url": "https://eulersolve.org/papers/owr-14299088-013/paper.pdf?v=f81117f10b49",
  "doi": "10.5281/zenodo.23049180",
  "zenodo_record_url": "https://zenodo.org/records/23049180",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Scope: the general degree-only bound is disproved for every degree bound at least 4. Degree bounds 2 and 3 remain open; the limitation proved for the elimination mechanism is not a solution for arbitrary pairs in those degrees. The construction builds on Basu and Perrucci's three-polynomial example. No absolute priority claim is made; the bounded literature search has explicitly recorded limitations. Self-audited, AI-assisted and unrefereed; no independent peer review is claimed.",
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    "source.zip": {
      "sha256": "a67e828f5a831bbd407d82d5188a4ac84038fcf7f857be9714d5b8cb28835080"
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    "verification_report.md": {
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}